Post on 18-Jan-2018
description
8-1: Exponential 8-1: Exponential GrowthGrowth
Objective CA 12: Students know the laws of fractional exponents, understanding exponential functions, and use these functions in problems
involving exponential growth and decay.
An exponential function involves the expression
where the base b is a positive number other than 1
xb
Graph the function ( ) 2xf x
xx--33
22-3 -3 ==
1/81/8--22
22-2 -2 = ¼= ¼
--11
22-1-1 = ½ = ½
00 2200 = 1 = 111 2211 = 2 = 222 2222 = 4 = 4
2xf x
Question:What do you notice about the x-values and the f(x) values?
4
2
f x = 2x
Notice the end behavior as , ( )As , ( ) , which means that the graph has the line y = 0 as an asymptote.
x f xx f x
An asymptote is a line that a graph approaches as you move away from the
origin.
Example 1: Graphing Example 1: Graphing Exponential Functions of the Exponential Functions of the
Form y = abForm y = abxx 2Graph the function: 23
xy
2 13 3Plot the points: (0, ), (1,1 )
4
2
-5
1, 3/2
0, 1/2 f x =
12
3x
23Graph the function: ( )xy
Plot the points: (0, 1), (1, 1.5)
-2
5
1, - 3/2 0, -1
f x = -32
x
To graph a general exponential function:
x hy ab k
Start by graphing . Then translate the graph by h units horizontally and k units vertically.
xy ab
Example 2: Graphing a Example 2: Graphing a General Exponential FunctionGeneral Exponential Function
1Graph: 3 2 4. State the domain and range.xy
Begin by sketching the graph 3 2xy
Plot :(0, 3)) and (1, 6). 10
5
f x = 32x
Translate the graph 1 unit to the right and down 4 units
8
6
4
2
-2
-5 5g x = 32x-1-4
f x = 32x
Homework:Homework:Page 469 #19-33 OddPage 469 #19-33 Odd
Table of Values needed for credit
Investigating Graph of Investigating Graph of Exponential Functions page Exponential Functions page
46546511. Graph 2 and 3 2 Compare the 3
graphs with the graph of 2 .12. Graph 2 and 5 2 . Compare the5
graphs with the graph of 2
3. Describe the effect of on the graph of 2whe
x x
x
x x
x
x
y y
y
y y
y
a y a
n is positive and when is negative.a a
4
2
-5 5
h x = 2x
g x = 32x
f x = 13
2x
1The graph of 2 lies below 2 and 3
has a y-intercept of 1/3, while 3 2 lies aboveand has a y-intercept of 3. All three graphs havethe same end behavior and same general slope.
x x
x
y y
y
2
-2
-4
-5 5
h x = 2x
g x = -52x f x = -15
2x
1The graph of 2 lies closer to the x-axis 5
than that of 2 so that y approaches the x-axis from below as , and
x
x
y
yx y
If 2 where reflected in the x-axis it would1lie between 2 and 5 2 .5
x
x x
y
y y
Describe the effect of a on the graph of 2when a is positive and when a is negative.
xy
If 0 1 then the graph of 2 lies below
2 while if 1 the graph of 2 lies
above the graph 2 . In either case, the graphhas the same end behavior and the same generalshape.
x
x x
x
a y a
y a y a
y
If 1 0 then the graph 2 lies closer
to the x-axis than the graph 2 but below thex-axis instead of above it.
If 1 the graph of 2 lies below the x-axis but grows away from the x-axis mo
x
x
x
a y a
y
a y a
re
quickly than that of 2xy
2xy a Note the following about the graph .
The graph passes through the point (0, a). The y-intercept is a.The x-axis is an asymptote of the graph.The domain is all real numbers.The range is y > 0 if a > 0 and y < 0 if a<0,
The characteristics of the graph 2 listed on
the previous slide are true of the graph of .
If 1 and 1, the function is an Exponential Growth Function.
x
x
x
y a
y ab
a b y ab